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Juan González-Meneses

Publications and source records attributed to Juan González-Meneses.

18 recordsLinked to original sources
arXiv2026

Classifying spaces for families of virtually abelian subgroups of surface braid groups

Given a group $G$ and an integer $n \geq 0$, let $\mathcal{F}_n$ denote the family of all virtually abelian subgroups of $G$ of rank at most $n$. In this article, we show that for each $n \geq 1$, the minimal dimension of a model for the classifying space $E_{\mathcal{F}_n}G$ for the pure braid group of a surface of non-negative Euler characteristic with at least one boundary component or one puncture is equal to the virtual cohomological dimension of $G$ plus $n$. We prove an analogous result for the full braid group of the sphere. As an application, we compute the minimal dimension of a model for the classifying space associated to the family of amenable subgroups of pure surface braid groups.

math.GR↗
arXiv2025

Parabolic subgroups of complex braid groups

In this paper we introduce a class of `parabolic' subgroups for the generalized braid group associated to an arbitrary irreducible complex reflection group, which maps onto the collection of parabolic subgroups of the reflection group. Except for one case, which is proven separately elsewhere, we prove that this collection forms a lattice, so that intersections of parabolic subgroups are parabolic subgroups. In particular, every element admits a parabolic closure, which is the smallest parabolic subgroup containing it. We furthermore prove that it provides a simplicial complex which generalizes the curve complex of the usual braid group. In the case of real reflection groups, this complex generalizes the one previously introduced by Cumplido, Gebhardt, González-Meneses and Wiest for Artin groups of spherical type. We show that it shares similar properties, and similarly conjecture its hyperbolicity, with a few additional results in this direction.

math.GR↗
arXiv2025

Conjugacy invariants and rigidity in Garside groups: a uniformity phenomenon

Consider an element~$x$ of a Garside group which is rigid in the sense of Garside-theory. Let $SC(x)$ be the set of rigid conjugates of~$x$ -- this is a well-known characteristic subset of the conjugacy class of~$x$. We present computational evidence that the sequence $( |SC(x^n)| )_{n\in\mathbb N}$ is not only bounded, but in fact periodic, and that there is a bound on the length of the period which depends only on the underlying group and its Garside structure. We prove this result in the special case of the circular Garside groups, including the 2-generator Artin groups with their classical and dual structures (where we prove that the sequence is always constant), and in the case of the dual $4$-strand braid group.

math.GR↗
arXiv2025

Canonical Reduction Systems in Artin-Tits groups of spherical type

We introduce the canonical reduction system of an element in an Artin-Tits group of spherical type, which generalizes the similar notion for braids (and mapping classes) introduced by Birman, Lubotzky and McCarthy. We show its basic properties, which coincide with those satisfied in braid groups, and we provide an algorithm to compute it. We improve the algorithm in the case of braid groups, and discuss its complexity in this case. As a necessary result for obtaining the general algorithm, we prove that the centralizers of positive powers of an element form a periodic sequence and we show how to compute its period.

math.GR↗
arXiv2025

Explicit formulae for the Aicardi-Juyumaya bracket of tied links

The double bracket $\langle \langle \cdot \rangle \rangle$ (also known as the AJ-bracket) is an invariant of framed tied links that extends the Kauffman bracket of classical links. Unlike the classical setting, little is known about the structure of AJ-states (analogous to classical Kauffman states) of a given tied link diagram, and no general state-sum formula for the AJ-bracket is currently available. In this paper we analyze the AJ-states of $2$- and $3$-tied link diagrams, and provide a complete description of their associated resolution trees leading to a computation of $\langle \langle \cdot \rangle \rangle$. As a result, we derive explicit state-sum formulas for the AJ-bracket. These are the first closed-form expressions of this kind, and they constitute a concrete step toward a combinatorial categorification of the tied Jones polynomial.

math.GT↗
arXiv2025

Positive 3-braids, Khovanov homology and Garside theory

Khovanov homology is a powerful invariant of oriented links that categorifies the Jones polynomial. Nevertheless, computing Khovanov homology of a given link remains challenging in general with current techniques. In this work we focus on links that are the closure of positive 3-braids. Starting with a classification of conjugacy classes of 3-braids arising from the Garside structure of braid groups, we compute, for any closed positive 3-braid, the first four columns (homological degree) and the three lowest rows (quantum degree) of the associated Khovanov homology table. Moreover, the number of rows and columns we can describe increases with the infimum of the positive braid (a Garside theoretical notion). We will show how to increase the infimum of a 3-braid to its maximal possible value by a conjugation, maximizing the number of cells in the Khovanov homology of its closure that can be determined, and show that this can be done in linear time.

math.GT↗
arXiv2020

Limits of sequences of pseudo-Anosov maps and of hyperbolic 3-manifolds

There are two objects naturally associated with a braid $β\in B_n$ of pseudo-Anosov type: a (relative) pseudo-Anosov homeomorphism $φ_β\colon S^2\to S^2$; and the finite volume complete hyperbolic structure on the 3-manifold $M_β$ obtained by excising the braid closure of $β$, together with its braid axis, from $S^3$. We show the disconnect between these objects, by exhibiting a family of braids $\{β_q:q\in{\mathbb{Q}}\cap(0,1/3]\}$ with the properties that: on the one hand, there is a fixed homeomorphism $φ_0\colon S^2\to S^2$ to which the (suitably normalized) homeomorphisms $φ_{β_{q}}$ converge as $q\to 0$; while on the other hand, there are infinitely many distinct hyperbolic 3-manifolds which arise as geometric limits of the form $\lim_{k\to\infty} M_{β_{q_k}}$, for sequences $q_k\to 0$.

math.GT↗
arXiv2019

The root extraction problem for generic braids

We show that, generically, finding the $k$-th root of a braid is very fast. More precisely, we provide an algorithm which, given a braid $x$ on $n$ strands and canonical length $l$, and an integer $k>1$, computes a $k$-th root of $x$, if it exists, or guarantees that such a root does not exist. The generic-case complexity of this algorithm is $O(l(l+n)n^3\log n)$. The non-generic cases are treated using a previously known algorithm by Sang-Jin Lee.

math.GR↗
arXiv2019

On parabolic subgroups of Artin-Tits groups of spherical type

We show that, in an Artin-Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to all Artin-Tits groups of spherical type a result that was previously known for braid groups. To obtain the above results, we show that every element in an Artin-Tits group of spherical type admits a unique minimal parabolic subgroup containing it. Also, the subgroup associated to an element coincides with the subgroup associated to any of its powers or roots. As a consequence, if an element belongs to a parabolic subgroup, all its roots belong to the same parabolic subgroup. We define the simplicial complex of irreducible parabolic subgroups, and we propose it as the analogue, in Artin-Tits groups of spherical type, of the celebrated complex of curves which is an important tool in braid groups, and more generally in mapping class groups. We conjecture that the complex of irreducible parabolic subgroups is $δ$-hyperbolic.

math.GR↗
arXiv2018

On lexicographic representatives in braid monoids

The language of maximal lexicographic representatives of elements in the positive braid monoid $A_n$ with $n$ generators is a regular language. We describe with great detail the smallest Finite State Automaton accepting such language, and study the proportion of elements of length $k$ whose maximal lexicographic representative finishes with the first generator. This proportion tends to some number $P_{n,1}$, as $k$ tends to infinity, and we show that $P_{n,1}\geq \frac{1}{8}$ for every $n\geq 1$. We also provide an explicit formula, based on the Fibonacci numbers, for the number of states of the automaton.

math.GR↗
arXiv2018

On the growth of Artin--Tits monoids and the partial theta function

We present a new procedure to determine the growth function of a homogeneous Garside monoid, with respect to the finite generating set formed by the atoms. In particular, we present a formula for the growth function of each Artin--Tits monoid of spherical type (hence of each braid monoid) with respect to the standard generators, as the inverse of the determinant of a very simple matrix. Using this approach, we show that the exponential growth rates of the Artin--Tits monoids of type $A_n$ (positive braid monoids) tend to $3.233636\ldots$ as $n$ tends to infinity. This number is well-known, as it is the growth rate of the coefficients of the only solution $x_0(y)=-(1+y+2y^2+4y^3+9y^4+\cdots)$ to the classical partial theta function. We also describe the sequence $1,1,2,4,9,\ldots$ formed by the coefficients of $-x_0(y)$, by showing that its $k$th term (the coefficient of $y^k$) is equal to the number of braids of length $k$, in the positive braid monoid $A_{\infty}$ on an infinite number of strands, whose maximal lexicographic representative starts with the first generator $a_1$. This is an unexpected connection between the partial theta function and the theory of braids.

math.GR↗
arXiv2017

Polynomial braid combing

Braid combing is a procedure defined by Emil Artin to solve the word problem in braid groups for the first time. It is well-known to have exponential complexity. In this paper, we use the theory of straight line programs to give a polynomial algorithm which performs braid combing. This procedure can be applied to braids on surfaces, providing the first algorithm (to our knowledge) which solves the word problem for braid groups on surfaces with boundary in polynomial time and space. In the case of surfaces without boundary, braid combing needs to use a section from the fundamental group of the surface to the braid group. Such a section was shown to exist by Gon\ccalves and Guaschi, who also gave a geometric description. We propose an algebraically simpler section, which we describe explicitly in terms of generators of the braid group, and we show why the above procedure to comb braids in polynomial time does not work in this case.

math.GT↗
arXiv2013

Closures of positive braids and the Morton-Franks-Williams inequality

We study the Morton-Franks-Williams inequality for closures of simple braids (also known as positive permutation braids). This allows to prove, in a simple way, that the set of simple braids is a orthonormal basis for the inner product of the Hecke algebra of the braid group defined by Kálmán, who first obtained this result by using an interesting connection with Contact Topology. We also introduce a new technique to study the Homflypt polynomial for closures of positive braids, namely resolution trees whose leaves are simple braids. In terms of these simple resolution trees, we characterize closed positive braids for which the Morton-Franks-Williams inequality is strict. In particular, we determine explicitly the positive braid words on three strands whose closures have braid index three.

math.GT↗
arXiv2012

Generating random braids

We present an algorithm to generate positive braids of a given length as words in Artin generators with a uniform probability. The complexity of this algorithm is polynomial in the number of strands and in the length of the generated braids. As a byproduct, we describe a finite state automaton accepting the language of lexicographically minimal representatives of positive braids that has the minimal possible number of states, and we prove that its number of states is exponential in the number of strands.

math.GR↗
arXiv2011

Twisted conjugacy in braid groups

In this note we solve the twisted conjugacy problem for braid groups, i.e. we propose an algorithm which, given two braids $u,v\in B_n$ and an automorphism $ϕ\in Aut (B_n)$, decides whether $v=(ϕ(x))^{-1}ux$ for some $x\in B_n$. As a corollary, we deduce that each group of the form $B_n \rtimes H$, a semidirect product of the braid group $B_n$ by a torsion-free hyperbolic group $H$, has solvable conjugacy problem.

math.GR↗
arXiv2008

The cyclic sliding operation in Garside groups

We present a new operation to be performed on elements in a Garside group, called cyclic sliding, which is introduced to replace the well known cycling and decycling operations. Cyclic sliding appears to be a more natural choice, simplifying the algorithms concerning conjugacy in Garside groups and having nicer theoretical properties. We show, in particular, that if a super summit element has conjugates which are 'rigid' (that is, which have a certain particularly simple structure), then the optimal way of obtaining such a rigid conjugate through conjugation by positive elements is given by iterated cyclic sliding.

math.GR↗
arXiv2008

Solving the conjugacy problem in Garside groups by cyclic sliding

We present a solution to the conjugacy decision problem and the conjugacy search problem in Garside groups, which is theoretically simpler than the usual one, with no loss of efficiency. This is done by replacing the well known cycling and decycling operations by a new one, called cyclic sliding, which appears to be a more natural choice. We give an analysis of the complexity of our algorithm in terms of fundamental operations with simple elements, so our analysis is valid for every Garside group. This paper intends to be self-contained, not requiring any previous knowledge of prior algorithms, and includes all the details for the algorithm to be implemented on a computer.

math.GR↗